how can the papers be arranged if: i) the 2mathematical papers are together and (ii) the 2mathematical papers are not consecutive.
For (i), we can treat the 2 papers as being one item (but they can be arranged in 2 ways), and we have to arrange 5 items. Let's call the math papers A and B so they can be arranged as A,B or B,A, and call the others C, D, E, F. So it can look like this for example:
(A, B) C, D, E, F, or C, D, (B, A), E, F
(B, A) C, D, E, F, or C, D, (A, B), E, F (these are just examples)
So we have to arrange five items, and then double that since we can also have A,B or B,A in each case.
So from 5 items we have to order/arrange 5 items, times 2 to double it, so find:
(ii) Notice that in part (i) we found the number of ways the papers can be consecutive, as they're consecutive when A and B are together. So now we want the number of ways the 6 papers can be arranged, with A and B not being consecutive... how do we find that?
We want to arrange all 6 papers this time, and subtract the number of ways that A, B can be consecutive - so subtract our answer from (i):
6P6 - 2(5P5)